Definition: Let V be a vector space. A subset W of V is called a subspace of V iff W is a vector space with respect to the operations of V, that is,
(S1) w1​+w2​∈W∀w1​,w2​∈W (closure under addition)
(S2) cw∈W∀c∈F and ∀w∈W (closure under scalar multiplication)
Remark: W is a subspace of V iff W is nonempty and W is closed under addition and scalar multiplication. All other axioms are inherited from the original vector space V.
Example: linear space V=R2, subspace W=[α 0]T∈R2:α∈R
Solution: Let w1​=[α1​ 0]T and w2​=[α2​ 0]T and c∈R,
(S1) w1​=[α1​0​] and w2​=[α2​0​] be two arbitrary elements of W. Then w1​+w2​=[α1​+α2​0​]∈W
(S2) cw1=[cα1​0​]∈W for all c∈R.
Hence W is a subspace of V.
Example: linear space V=R2, subspace W=[α 1]T∈R2:α∈R
Solution: Let w1​=[α1​ 1]T and w2​=[α2​ 1]T
(S1) w1​=[α1​1​] and w2​=[α2​1​] be two arbitrary elements of W. Then w1​+w2​=[α1​+α2​2​]∈/W
Remark: In R2, a subspace is a line through the origin. Any line through the origin is a subspace of R2.
In function spaces, an example can be given as follows:
Example: linear space V= set of all real-valued functions of a real variable t→f(t);
subspace W1​= set of all continuous functions [+]
subspace W2​= set of all constant functions [+]
subspace W3​= set of all functions periodic with π [+]
subspace W4​= set of all functions which are discontinuous at t=1 [-]
Remark: 0 vector is a subspace of any vector space, even subspace of itself, and it is the smallest subspace.
[+] W1​,W2​,W3​ are subspaces of V W4​ is not a subspace of V
Example: Show that Y+Z is a linear subspace of X, if Y and Z are also linear subspaces of X.
Proof: Let w1​+w2​∈W, with w1​=y1​+z1​ where y1​∈Y, z1​∈Z w2​=y2​+z2​ where y2​∈Y, z2​∈Z
then
Example: For Y and Z are subspaces of X, show that whether Y∪Z is a subspace of X or not.  Proof:
Prove by contradiction.
Assume Y∪Z is a subspace of X. Then Y∪Z is closed under addition and scalar multiplication.
Let Y={(y,0):y∈R} and Z={(0,z):z∈R}.
Then u1​=(1,0)∈Y and u2​=(0,1)∈Z.
u1​+u2​=(1,1)∈/Y∪Z.
Example: Is R2 a subspace of complex vector space C2?  Proof: Note that we consider complex vector space, so if x∈R2 then x∈C2.
i∈C and x∈R2 then ix∈R2.
i(1,1)=(i,i)∈/R2
Hence R2 is not a subspace of C2.
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Subspaces
Sums of Subspaces
Definition: Suppose W1​,...,Wm​ are subspaces of a vector space V. The sum of W1​,...,Wm​ is the set of all possible sums of elements of W1​,...,Wm​:
Remark: W1​+...+Wm​ is the smallest subspace of V containing W1​,...,Wm​.
Direct Sums
Definition: Suppose W1​,...,Wm​ are subspaces of a vector space V. The sum W1​+...+Wm​ is called a direct sum if each element of W1​+...+Wm​ can be written in one and only one way as a sum w1​+...+wm​ with wi​∈Wi​.
Remark: if W1​+...+Wm​ is a direct sum, then W1​⊕...⊕Wm​ is used to denote the direct sum.
Example: Suppose Uj​ is a subspace of Fn of those vectors whose jth component is zero. As U2​={(0,x,...,0)∈Fn:x∈F}, Then.